When proving the Thom's Cobordism theorem for unoriented manifolds, at some point we are able to create a map $$\phi: MO\rightarrow \bigvee_{i}\Sigma^{|d_i|}K(\mathbb{Z_2})$$ where $MO$ is the Thom spectrum and $K(\mathbb{Z_2})$ is the Eilenberg-Maclane spectrum, such that it gives an isomorphism in $\mathbb{Z}_2$ cohomology.

Following these notes http://math.uchicago.edu/~may/REU2018/REUPapers/Slaoui.pdf then the author claims that since $\pi_*(MO)$ is $2$-torsion, which can be seens from the fact that it's the same as the cobordism ring then the map gives an isomorphism in integral homology. I was able to see why this is true if we were dealing with Topological spaces , we can use the Hurewicz theorem for rational coefficients and localizations to get the desired result. However I am not sure that these arguments generalize to spectra ? Maybe from the fact that $\pi_*(MO)$ is bounded from below we do have that this is true , however I am not sure.

Any insight is appreciated, thanks in advance.


1 Answer 1


It is indeed true that if $f\colon X\to Y$ is a map between bounded below spectra induces an isomorphism in integral homology, then it is already a weak equivalence. This amounts to showing that the homotopy cofiber of $f$ (which is also bounded below by the long exact sequence) is (weakly) contractible. Note that $H_\ast(X;\mathbb{Z}) = \pi_\ast(X\wedge H\mathbb{Z})$ and that the smash product of spectra commutes with all (homotopy) colimits, in particular smashing with $H\mathbb{Z}$ preserves cofiber sequences, hence we need to show that if $X$ is a bounded below spectrum with vanishing integral homology, then $X$ is contractible. To see why this is true, suppose $X$ is not contractible and that $n$ is minimal such that $\pi_n(X)\neq 0$. It is a fact that the Hurewicz map $\pi_n(X)\to H_n(X;\mathbb{Z})$ in the case at hand, but we assumed that the homology of $X$ vanished, thus we obtained a contradiction.

Next, we want to see why your map $\phi$ induces an isomorphism in integral homology. Since both source and target are $2$-torsion, their rational homology vanishes and it is shown in the reference you gave that $\phi$ also induces an isomorphism in homology with coefficients in $\mathbb{Z}/p^\infty$ for all primes $p$ (by the way, this also implicitly uses that the smash product commutes with homotopy colimits), hence also in homology with coefficients in $\bigoplus_p \mathbb{Z}/p^\infty$, since the smash product commutes with direct sums. The short exact sequence $$\mathbb{Z}\to\mathbb{Q}\to \bigoplus_p \mathbb{Z}/p^\infty$$ give rise to a cofiber sequence of spectra $$H\mathbb{Z}\to H\mathbb{Q}\to \bigoplus_p H\mathbb{Z}/p^\infty$$ and smashing this cofiber sequence with $\phi$ gives a map between cofiber sequences which is an equivalence on final two terms, hence also $\varphi\wedge H\mathbb{Z}$ is an equivalence.


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